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Physical Review C

Stochastic one-body transport and coupling to mean-field fluctuations

Sakir Ayik B, Yasuhisa Abe

Published Jul 3, 2001• 7 citations
Publisher

Abstract

A stochastic transport description for the single-particle density matrix is briefly discussed. It is shown that the stochastic description contains, in addition to incoherent binary collisions, a coherent damping mechanism due to coupling between mean-field fluctuations and single-particle motion, and an expression for the coherent collision term is derived. In the limit of small fluctuations around equilibrium, the collective and single-particle self-energies due to the coherent mechanism are deduced.

DOI
10.1103/physrevc.64.024609

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A stochastic transport description for the single-particle density matrix is briefly discussed. It is shown that the stochastic description contains, in addition to incoherent binary collisions, a coherent damping mechanism due to coupling between mean-field fluctuations and single-particle motion, and an expression for the coherent collision term is derived. In the limit of small fluctuations around equilibrium, the collective and single-particle self-energies due to the coherent mechanism are deduced.
We present a formal derivation of the exact dynamics of the one-body density matrix. Its essential ingredients are shown to be the following: (a) a mean field unitary time evolution; (b) irreducible nonunitary corrections to the unitary evolution (collision effects); and (c) effects due to the time evolution of initial state correlation [which contribute to both (a) and (b)]. In the weak coupling limit the collision effects give rise to a term of Boltzmann form. A qualitative discussion of the importance of collision effects for the expectation values of one-body operators and a quantitative illustration in the framework of an exactly soluble model are given. In this case one finds large nonunitary contributions.NUCLEAR STRUCTURE Exact microscopic description of the time evolution of the one body density matrix.
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.
Starting from a recent reformulation of the Markov limit, we apply the mean- field approximation to the resulting Lindblad-type many-electron dynamics, and derive a closed equation of motion for the electronic single-particle density matrix in the presence of one- and two-body scattering mechanisms. The proposed formulation preserves the positive- definite character of the single-particle density matrix. This result is in striking contrast with conventional Markov approaches, where the single-particle mean-field equations can lead to positivity violations and therefore to unphysical results.
Assuming that the effect of the residual interaction beyond the mean field is weak and has a short memory time, two approximate treatments of correlation in fermionic systems by means of the Markovian quantum jump are presented. A simplified scenario for the introduction of fluctuations beyond the mean field is presented first. In this theory, part of the quantum correlations between the residual interaction and the one-body density matrix are neglected and jumps occur between many-body densities formed of pairs of states $D=|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|/\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$, where $|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$ and $|{\ensuremath{\Phi}}_{b}\ensuremath{\rangle}$ are antisymmetrized products of single-particle states. The underlying stochastic mean-field theory is discussed and is applied to the monopole vibration of a spherical 40Ca^{40}\mathrm{Ca}40Ca nucleus under the influence of a statistical ensemble of two-body contact interactions. This framework is however too simplistic to account for both fluctuation and dissipation. In the second part of this work, an alternative quantum jump method is obtained without making the approximation on quantum correlations. By restricting to two-particle--two-hole residual interactions, the evolution of the one-body density matrix of a correlated system is transformed into a Lindblad equation. The associated dissipative dynamics can be simulated by quantum jumps between densities written as $D=|\ensuremath{\Phi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\Phi}|$, where $|\ensuremath{\Phi}\ensuremath{\rangle}$ is a normalized Slater determinant. The associated stochastic Schr\"odinger equation for single-particle wave functions is given.
The nonequilibrium density matrix technique is utilized to study the incoherent transfer of a quantum particle coupled to a fluctuating environment. The dynamics of the environment is assumed to split into two different types of molecular motions. The first type is associated with low-amplitude vibrations forming a thermal bath (TB). High-amplitude displacements of separated molecular groups define the second type and create stochastic fields for acting on the transferred particle. Specifying the TB by a set of harmonic oscillators and the stochastic fields by a dichotomic process averaged kinetic equations were derived. These equations enable us to construct a rate constants expressions and to find the steady-state populations for a quantum particle. Regimes of enhanced or reduced particle transfer are discussed in detail in relation to the characteristics of the stochastic field and the TB. \textcopyright{} 1996 The American Physical Society.